Average Error: 10.3 → 1.5
Time: 5.3s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\left(\frac{y}{a - z} - \frac{z}{a - z}\right) \cdot t + x\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\left(\frac{y}{a - z} - \frac{z}{a - z}\right) \cdot t + x
double code(double x, double y, double z, double t, double a) {
	return (x + (((y - z) * t) / (a - z)));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((((y / (a - z)) - (z / (a - z))) * t) + x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.3
Target0.5
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 10.3

    \[x + \frac{\left(y - z\right) \cdot t}{a - z}\]

  2. Simplified1.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)}\]
  3. Using strategy rm

  4. Applied fma-udef1.5

    \[\leadsto \color{blue}{\frac{y - z}{a - z} \cdot t + x}\]
  5. Using strategy rm

  6. Applied div-sub1.5

    \[\leadsto \color{blue}{\left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \cdot t + x\]

  7. Final simplification1.5

    \[\leadsto \left(\frac{y}{a - z} - \frac{z}{a - z}\right) \cdot t + x\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

  (+ x (/ (* (- y z) t) (- a z))))